OAI.FiniteFactor.exists_source_prepared_finite: intermediate lemma from OpenAI, Invariant-projection counterexamples for every irrational rotation

Proved

OAI.FiniteFactor.exists_source_prepared_finite

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.FiniteFactor.main_theorem, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Invariant-projection counterexamples for every irrational rotation, OpenAI Math Release, September 27, 2026, result family 293, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.FiniteFactor.exists_source_prepared_finite of lean/OAI/Analysis/FiniteFactor/CrossedProduct/PreparedFinite.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.FiniteFactor, with open MeasureTheory Filter, open scoped ENNReal Topology):

theorem exists_source_prepared_finite : ∃ θ : SourceCoordinate → ℝ,
    SourceRationalIndependent θ ∧
    (∃ j : ℕ, SourceSecondPhasesFrom θ j) ∧
    SourceOrderedTorusSampling θ ∧
    (∀ k : ℤ, MeasurePreserving (coordinateRotate θ k) sourceSamplingHaar sourceSamplingHaar) ∧
    (∀ k : ℤ, k ≠ 0 → ∀ x, coordinateRotate θ k x ≠ x) ∧
    IsFiniteAlgebra (sourceAlgebra θ) ∧
    Nonempty (TracialRepresentation (sourceAlgebra θ)) ∧
    ∃ hθ : SourceAngles θ,
      let T := sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg
      T ∈ sourceAlgebra θ ∧ T ≠ 0 ∧ Function.Injective T ∧ ‖T‖ ≤ 1 ∧
        Tendsto (fun n : ℕ => ‖T^n‖^(1/(n : ℝ))) atTop (𝓝 0)

Definitions. The statement is written with OAI.FiniteFactor.SourceAngles, OAI.FiniteFactor.SourceCoordinate, OAI.FiniteFactor.SourceHilbert, OAI.FiniteFactor.SourceOrderedTorusSampling, OAI.FiniteFactor.SourceRationalIndependent, OAI.FiniteFactor.SourceSecondPhasesFrom, OAI.FiniteFactor.TracialRepresentation, OAI.FiniteFactor.coordinateRotate, OAI.FiniteFactor.sourceAlgebra, OAI.FiniteFactor.sourceBlockSize, OAI.FiniteFactor.sourceLogWeight, OAI.FiniteFactor.sourceMeasure, OAI.FiniteFactor.sourceSamplingHaar and OAI.FiniteFactor.SourceAngles.nonneg (definitions bundle OAIQ3Ff000 of this split); OAI.FiniteFactor.sourceLogWeight_nonneg and OAI.FiniteFactor.sourceOperator (definitions bundle OAIQ3Ff002 of this split); OAI.FiniteFactor.IsFiniteAlgebra (the published bundle Def_FiniteFactor).

Used by. The proof of OAI.FiniteFactor.main_theorem of this split imports it.

Preamble

import Mathlib
import Definitions.Def_FiniteFactor
import Definitions.Def_OAIQ3Ff002

Formal statement

section

attribute [-instance] _root_.OAI.FiniteFactor.sourceBiasedRealMeasure_probability _root_.OAI.FiniteFactor.sourceFiniteReferenceKernel_markov _root_.OAI.FiniteFactor.sourceNormalFunctionalSpace_complete _root_.OAI.FiniteFactor.sourceNormalPredualSpace_complete _root_.OAI.FiniteFactor.sourceNormalPredualSpace_normedGroup _root_.OAI.FiniteFactor.sourceNormalPredualSpace_normedSpace _root_.OAI.FiniteFactor.sourceResamplingKernel_markov
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal Topology
noncomputable section

theorem exists_source_prepared_finite : ∃ θ : SourceCoordinate → ℝ,
    SourceRationalIndependent θ ∧
    (∃ j : ℕ, SourceSecondPhasesFrom θ j) ∧
    SourceOrderedTorusSampling θ ∧
    (∀ k : ℤ, MeasurePreserving (coordinateRotate θ k) sourceSamplingHaar sourceSamplingHaar) ∧
    (∀ k : ℤ, k ≠ 0 → ∀ x, coordinateRotate θ k x ≠ x) ∧
    IsFiniteAlgebra (sourceAlgebra θ) ∧
    Nonempty (TracialRepresentation (sourceAlgebra θ)) ∧
    ∃ hθ : SourceAngles θ,
      let T := sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg
      T ∈ sourceAlgebra θ ∧ T ≠ 0 ∧ Function.Injective T ∧ ‖T‖ ≤ 1 ∧
        Tendsto (fun n : ℕ => ‖T^n‖^(1/(n : ℝ))) atTop (𝓝 0) := by
  sorry

end
end FiniteFactor

end OAI
end

Source

OpenAI, Invariant-projection counterexamples for every irrational rotation, OpenAI Math Release, September 27, 2026, https://github.com/openai/math (result family 293; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/FiniteFactor/CrossedProduct/PreparedFinite.lean (declaration OAI.FiniteFactor.exists_source_prepared_finite)

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